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Algebraic characterization of the isometries of the hyperbolic 5-space. (English) Zbl 1190.37022
The author obtains an algebraic characterization of the dynamical types of the orientation preserving isometries of the hyperbolic 5-space ${ℍ}^{5}$. The characterization is described by using the representation of the isometries of ${ℍ}^{5}$ as $2×2$ matrices over the quaternions, $𝔾𝕃\left(2,ℍ\right)$, together with the natural embedding of $𝔾𝕃\left(2,ℍ\right)$ into $𝔾𝕃\left(4,ℂ\right)$. It is also given the determination of the conjugacy classes and the $z$-classes in $𝔾𝕃\left(2,ℍ\right)$.
##### MSC:
 37C85 Dynamics of group actions other than $ℤ$ and $ℝ$, and foliations 51M10 Hyperbolic and elliptic geometries (general) and generalizations 15A33 Matrices over special rings
##### References:
 [1] Ahlfors L.V.: Möbius Transformations and Clifford Numbers. Differential Geometry and Complex Analysis, pp. 65–73. Springer, New York (1985) [2] Aslaksen H.: Quaternionic determinants. Math. Intell. 18, 57–65 (1996) · Zbl 0881.15007 · doi:10.1007/BF03024312 [3] Beardon A.F.: The Geometry of Discrete Groups. Graduate Texts in Mathematics 91. Springer- Verlag, Berlin (1983) [4] Brenner J.L.: Matrices of quaternions. Pacific J. Math. 1, 329–335 (1951) [5] Cao W., Parker J.R., Wang X.: On the classification of quaternionic Möbius transformations. Math. Proc. Cambridge Philos. Soc. 137(2), 349–361 (2004) · Zbl 1059.30043 · doi:10.1017/S0305004104007868 [6] Cao, C. Waterman, P.L.: Conjugacy Invariants of Möbius Groups: Quasiconformal Mappings and Analysis pp. 109–139. Springer (1998) [7] Cao W.: On the classification of four-dimensional Möbius transformations. Proc. Edinb. Math. Soc. 50(1): 49–62 (2007) · Zbl 1121.30022 · doi:10.1017/S0013091505000398 [8] Foreman B.: Conjugacy invariants of $𝕊L\left(2,ℍ\right)$ . Linear Algebra Appl. 381, 25–35 (2004) · Zbl 1048.15015 · doi:10.1016/j.laa.2003.11.002 [9] Gongopadhyay K., Kulkarni R.S.: z-Classes of isometries of the hyperbolic space. Conform. Geom. Dyn. 13, 91–109 (2009) · Zbl 1206.51017 · doi:10.1090/S1088-4173-09-00190-8 [10] Gongopadhyay, K.: z-Classes of isometries of pseudo-riemannian geometries of constant curvature. Ph.D. Thesis, IIT Bombay (2008) [11] Gongopadhyay, K.: Dynamical types of isometries of hyperbolic space of dimension 5. arXiv:math/0511444v1 [12] Kellerhals R.: Collars in $PSL\left(2,ℍ\right)$ . Ann. Acad. Sci. Fenn. Math. 26, 51–72 (2001) [13] Kellerhals R.: Quaternions and some global properties of hyperbolic 5-manifolds. Canad. J. Math. 55, 1080–1099 (2003) · Zbl 1054.57019 · doi:10.4153/CJM-2003-042-4 [14] Kulkarni R.S.: Dynamical types and conjugacy classes of centralizers in groups. J. Ramanujan Math. Soc. 22(1), 35–56 (2007) [15] Kulkarni R.S.: Dynamics of linear and affine maps. Asian J. Math. 12(3), 321–344 (2008) [16] Lee H.C.: Eigenvalues and canonical forms of matrices with quaternion coefficients. Proc. Roy. Irish Acad. Sect. A. 52, 253–260 (1949) [17] Parker J.R., Short I.: Conjugacy classification of quaternionic Möbius transformations. Comput. Methods Funct. Theory 9, 13–25 (2009) [18] Ratcliffe J.G.: Foundation of Hyperbolic Manifolds. Graduate Texts in Mathematics 149. Springer, New York (1994) [19] Waterman P.L.: Möbius groups in several dimensions. Adv. Math. 101, 87–113 (1993) · Zbl 0793.15019 · doi:10.1006/aima.1993.1043 [20] Wilker J.B.: The quaternion formalism for Möbius groups in four or fewer dimensions. Linear Algebra Appl. 190, 99–136 (1993) · Zbl 0786.51005 · doi:10.1016/0024-3795(93)90222-A